■正多面体の正多角形断面(その53)

 8点

  (1,0,0,0,0,0,0,0)

  (0,1,0,0,0,0,0,0)

  (0,0,1,0,0,0,0,0)

  (0,0,0,1,0,0,0,0)

  (0,0,0,0,1,0,0,0)

  (0,0,0,0,0,1,0,0)

  (0,0,0,0,0,0,1,0)

  (0,0,0,0,0,0,0,1)

が,xy平面上の8点

  (cos0π/8,sin0π/8)

  (cos2π/8,sin2π/8)

  (cos4π/8,sin4π/8)

  (cos6π/8,sin6π/8)

  (cos8π/8,sin8π/8)

  (cos10π/8,sin10π/8)

  (cos12π/8,sin12π/8)

  (cos12π/8,sin12π/8)

に投影されるためには,2×8行列

M=[cos0π/8,cos2π/8,・・・,cos14π/8]

  [sin0π/8,sin2π/8,・・・,sin14π/7]

が必要になる.

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正八角形の場合はX=1+2cos(360/8)=1+√2

x6=1/(4X+4)

x5=X/(4X+4)

x4={X+1}/(4X+4)

x3={X+1}/(4X+4)

x2={X}/(4X+4)

x1=(1)/(4X+4)

X0=x1+x2・cosθ+x3・cos2θ+x4・cos3θ+x5・cos4θ+x6・cos5θ

=x1+x2・cosθ+x3・cos2θ+x3・cos3θ+x2・cos4θ+x1・cos3θ

=x1-x2+x2・cosθ+x3・cos3θ+x1・cos3θ

=x1-x2+(x2-x3-x1)√2/2

Y0=x2・sinθ+x3・sin2θ+x4・sin3θ+x5・sin4θ+x6・sin5θ

=x2・sinθ+x3・sin2θ+x3・sin3θ+x2・sin4θ+x1・sin5θ

=x2・sinθ+x3+x3・sin3θ-x1・sin3θ

=x3+x2・sinθ+x3・sin3θ-x1・sin3θ

=x3+(x2+x3-x1)√2/2

X0^2+Y0^2を求めることになる。

=(x1-x2)^2+(x2-x3-x1)^2/2+(x1-x2)(x2-x3-x1)√2

+(x3)^2+(x2+x3-x1)^2/2+x3(x2+x3-x1)√2

=(1-x)^2+(-2)^2/2-2(1-x)√2

+(x+1)^2+(2X)^2/2+2x(x+1)√2・・・/(4x+4)^2

=(2x^2+2)+(4x^2+4)/2+(2x^2+4x-2)√2

={2(2x^2+2)+(2x^2+4x-2)√2}//(4x+4)^2

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